-Rn -f(p) p0- . -f p0, p0 {pn}, pn≠p0, - {f(pn)} . : limf(p)=a p p0
28. - . -f(p), ϵRn, , .
-f(p), ϵRn p0ϵ, limf(p)=f(p0) pp0
30. - - , .
Zx=lim∆xZ/∆x=lim(f(x0+∆x,y0)-f(x0,y0))/∆x ∆
Zy=lim∆yZ/∆y=lim(f(x0,y0+∆y)-f(x0,y0))/∆y ∆y
31. -z=f(x,y) (x0,y0), : ∆z=f(x,y)-f(x0,y0)=fx(x0,y0)∆x+ fy(x0,y0)∆y+eρ, ∆z=dz+ eρ, =(∆x,∆y)- - ∆x→0,∆y→0; ρ=√((∆x)2+∆y2)- (x,y) (x0,y0)
32. . - - : dz=zx∆x+zy∆y
33. . fx(x,y) fy(x,y) (x0,y0), -z=f(x,y)
34. . -z=f(x,y) (x0,y0), .
35. . - z(x;y) α, (;) t z(tx;ty)= tα z(x;y).
D Rn Rn, (x1, x2, ., xn) (tx1, tx2, ., txn) t>0 - f(x1, x2, ., xn) D λ, t>0 f (tx1, tx2, ., txn)=tλ f(x1, x2, ., xn).
36. . fx(tx, ty)x+fy(tx, ty)y=λtλ-1f(x,y)
t=1, :
fx(x, y)x+fy(x, y)y=λf(x,y)
38. . -f(x,y) (x0,y0) ( ) : (δf(x0,y0))/δ↑=lim ((f(x0 +tex, y0+tey)-f(x0,y0))/t) t→0+0
39. . - z= f(x,y) M(x,y) , , M(x,y).
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Grad f(M)=(fx(M),fy(M)). -, .
│Gradf(M)│=δf(M)/δe
ѱ(t)=f(p+tv), p,vϵRn, ѱ(0)=(gradf(p),v)
-gradf(M)
-f(x,y,z) -
41. . zxyzyx M(x0,y0) b , zxy=zyx
43. . ↑ () -f(x↑), - Ue(a↑)={x↑ϵRn:│x↑-a↑│<e} ↑, ↑ϵUe(a↑) f(x↑)≤f(a↑) (f(x↑)≥f(a↑)). .
44. . , -f(x↑) ↑, 0. -f(x↑) ↑ , : ↑- , d2fa↑ . ( )
40. . fx(x,y) fy(x,y) -f(x,y). -. zxyzyx .
45. . -n f(x↑) ↑ , : d2fa↑ , ↑- f ( ).
-f(x,y) P. ∆=detf(P)=fxx(P) fyy(P)-(fxy(P))2 ∆>0, - , fxx(P)<0- , fxx(P)>0- ; ∆<0,
46. . *↑ϵ () -f, ↑ϵ : f(x↑)≤ f(x*↑) f(x↑)≥f(x*↑)-
47. : : f(x,y) g(x,y).
1) L(x,y,λ)= f(x,y)+λg(x,y).
2) L/x=0
L/=0 (*; *)
L/λ=0
3) ) (*; *) - , (1; 1), g(x1;y1)=0 f(x*;y*) f(1; 1) , (*; *) - max min
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) , -f , min, max.
48. . - , , .
50. : -f(x,y) G x [a,b] ⌠g1(x)g2(x)f(x,y)dxdy=⌠ba{⌠g1(x)g2(x)f(x,y)dy}dx
55. - . . . S1=a1, S2=a1 + a2, Sn=a1+a2++an 1+2+3+.++.= . .. , -S1, S2, ., Sn, .. 1+2+3+.++.= , -S1, S2, ., Sn, .. S 1+2+3+.++.= . .
57. . , . . S. n Sn=Sn-1+an, an=Sn-Sn-1. n→∞ Sn Sn-1 S, an=Sn-Sn-1 , limn→∞an= limn→∞Sn- limn→∞Sn-1=S-S=0. , , , .. . .
58. . , >0 n=1,2,.... , , - .
59. . , , .
60. , : a1+a2+.+an+. q<1, n(, n) an+1/an<q, . an+1/an>1 n, .
: limn→∞an+1/an=d, d<1, d>1.
: -y=f(x) x>1. f(1)+f(2)++f(n)+ , ∫1∞f(x)dx
: : a1+a2+.+an+. b1+b2+.+bn+., : an<bn (n= 1, 2,.). () (). .
: a1+a2+.+an+. b1+b2+.+bn+., limn→∞an/bn=u, a1+a2+.+an+. b1+b2+.+bn+. .
61. . . , .. . : 1-2+3-4+.+(-1)-1+, . a1+a2+.+an+. , , , , . , .. , .
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62. . , n→∞, 1) 2) .
64. . a0+a1x+a2x++anxn+ =0, , , , , ||<|x0|; a0+a1x+a2x++anxn+ =1, , ||>|x1|
65. . : a0+a1x+a2x++anxn+ : 1) =0; 2) ; 3) R>0, (-R;R) [-R;R].
: (-R;R) a0+a1x+a2x++anxn+, R .
66. . - (-R;R) f(x)=a0+ a1x+ a2x2+..+ anxn+..(1) a1+ 2a2x+..+nanxn-1+..(2) :
1) (2) R, (1)
2) (-R,R) -f(x) f(x), (2)
: -f(x), (1) (-R,R), . (1). .
-f(x) (-R,R), . : ⌠x2x1f(x)dx=⌠x2x1a0+ a1x+ a2x2+..+ anxn+.dx=⌠x2x1 a0+⌠x2x1 a2 x2dx+⌠x2x1 anxndx+..
69. ex=1+x+(x2/2!)+(x3/3!)++(xn/n!)+
Sinx=x-(x3/3!)+(x5/5!)-+(-1)n(x2n+1/(2n+1)!)+
Cosx=1-(x2/2!)+(x4/4!)-+(-1)n(x2n/(2n)!)
1/1+x=1-x+x2-x3++(-1)nxn+(r=1)
Ln(1+x)=x-x2/2+x3/3-x4/4++(-1)n(xn+1/n+1)
(1+x)b=1+b/1!x+(b(b-1)x2)/2!++(b(b-1)(b-n+1)xn)/n!+
63. . a0+a1x+a2x++anxn+, a0, a1, a2, ,an, - -, .
70. . (0;0) -f(x,y) , fy, (0;0), y=f(x,y), y(x0)=y0 , .
71. . . , . : y=f(x)g(y), f(x) g(y) -.
. : y=g(y).
72. . y+p(x)y=0 , y+p(x)y=f(x).
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73. . N(x,y)dx+M(x,y)dy=0, N(x,y) M(x,y) D-, , -U(x,y), dU= N(x,y)dx+M(x,y)dy
75. . . n- , . yn+a1xyn-1+a2xyn-2++anxy=f(x), a1x, a2x, , anx, f(x) -.
W(y1, y2,,yk)=|y1 y2 . Yk|
|y1y2 . Yk|
|.. |
|y1(k-1)y2(k-1) . Yk(k-1)|
y1(x),., yn(x), n L(y)=0, .
74. . . y+p(x)y=f(x)yn (n≠0, n≠1) .
76. ( ). 1 : y=C1eλx+C2λx2 : =(1cosβx+C2sinβx) 3 : y=eλx(C1+C2x)
77. . y=xleax(Pm(x)cosbx+Qm(x)sinbx)